Reading a Graph
A graph turns an equation into a shape, and the shape answers questions the algebra hides. Where the curve crosses the x-axis are the roots; where it crosses the y-axis is the value at x = 0; where two curves meet is the solution to the system.
Writing Expressions
| To write | Type |
|---|---|
| x squared | x^2 |
| 3 times x | 3*x or 3x |
| square root | sqrt(x) |
| natural log | ln(x) |
| log base 10 | log(x) |
| e to the x | exp(x) |
| absolute value | abs(x) |
| trigonometry | sin(x), cos(x), tan(x) — in radians |
Constants pi and e are recognised. Implicit multiplication works, so 2x and 3(x+1) parse correctly.
Shapes Worth Recognising
- y = mx + b — a straight line, slope m, intercept b.
- y = ax² + bx + c — a parabola, opening up when a > 0.
- y = aⁿ — exponential growth or decay, never touching the axis.
- y = 1/x — a hyperbola with asymptotes on both axes.
- y = sin(x) — a wave repeating every 2π.
- y = √x — half a sideways parabola, undefined for negative x.
Choosing the Window
The visible range determines what you see, and a badly chosen one hides everything interesting. A parabola with roots at 200 and 210 looks like a vertical line in a −10 to 10 window. If a graph looks blank or featureless, widen the range first, then narrow it around whatever appears.
Where Numerical Plotting Struggles
Any plotter samples the function at finite points and joins them. Near a vertical asymptote — as in 1/x at zero — that produces a spurious near-vertical line joining a large positive value to a large negative one. It is a drawing artefact, not part of the function.
Frequently Asked Questions
Why does tan(x) look broken?
Because it genuinely is discontinuous, jumping to infinity every π radians. The near-vertical connecting lines are the plotter joining points across a gap that has no curve in it.
Are the trig functions in degrees or radians?
Radians, which is the mathematical standard. To plot in degrees, write sin(x*pi/180).
How do I find where two curves intersect?
Plot both and read the crossing point, then refine it by subtracting one from the other and finding where that difference is zero — which the root-finding in the results table does automatically.